# Continuity 2

## One-sided continuity

• One-sided continuity is defined similarly to two-sided continuity.
• Let $$a$$ be a number and let $$f(x)$$ be a function.
• If we have $$\lim_{x\to a^-}f(x)=f(a)$$, then we say that $$f(x)$$ is continuous at $$a$$ from the left.
• If we have $$\lim_{x\to a^+}f(x)=f(a)$$, then we say that $$f(x)$$ is continuous at $$a$$ from the right.
• For example, consider the function $$f(x)=[[x]]$$.
• If $$a$$ is not an integer, then we have $$\lim_{x\to a^-}f(x)=[[a]]=\lim_{x\to a^+}f(x)$$, therefore $$f(x)$$ is continuous at $$a$$.
• If $$a$$ is an integer, then we have $$\lim_{x\to a^-}f(x)=[[a]]-1$$ and $$\lim_{x\to a^+}f(x)=[[a]]$$. Therefore, $$f(x)$$ is continuous at $$a$$ from the right.
• Exercise. 2.5.42.

## Continuity on an interval

• Let $$f(x)$$ be a function and $$I$$ an interval. Then $$f(x)$$ is continuous on $$I$$, if it is continuous at each point of $$I$$.
• If an endpoint $$a$$ of $$I$$ is closed, and $$f(x)$$ is only defined on the side of $$a$$ in the direction of $$I$$, then instead of requiring two-sided continuity at $$a$$, we only require that $$f(x)$$ is continuous at $$a$$ from the direction of $$I$$.
• For example, let $$f(x)=1-\sqrt{1-x^2}$$ and $$I=[-1,1]$$.
• For $$-1<a<1$$, we require two-sided continuity: $\lim_{x\to a}f(x)=\lim_{x\to a}(1-\sqrt{1-x^2})=1-\lim_{x\to a}\sqrt{1-x^2}=1-\sqrt{1-a^2}=f(a).$
• Since the domain of $$f(x)=1-\sqrt{1-x^2}$$ is $$[-1,1]=I$$, in case $$a=\pm1$$, we only require continuity from the direction of $$I$$: $\lim_{x\to(-1)^+}f(x)=1=f(-1),\quad\lim_{x\to1^-}f(x)=1=f(1).$
• All this shows that $$f(x)$$ is continuous on $$I$$.
• Note that the graph of $$f(x)$$ is the lower half of the circle $x^2+(y-1)^2=1.$
• We say that $$f(x)$$ is continuous everywhere, if it is continuous on the entire real line $$\mathbf R=(-\infty,\infty)$$.